1.1 Quantum States

Quantum states are described by vectors in a complex vector space, specifically a Hilbert space, which is a complete inner product space. The exact properties of the Hilbert space are determined by the nature of the physical problem we are studying. Physically distinct states of the system are represented by orthogonal directions in Hilbert space. We will discuss what this means more rigorously in later chapters. A Hilbert space may be finite-dimensional, where the system can only be in finitely-many states, or infinite-dimensional, which is certainly the case for any particle that can move.

Transformations on the state of the system, such as time-evolution or change of basis, are described using linear operators, which are a generalisation of matrices from finite-dimensional linear algebra.

1.1.1 Dirac Notation

States in the Hilbert space are written as $\lvert\psi\rangle$ , where $\psi$ is some label we have made up for the state. This notation is called a ket. Recall that the inner product can be defined as the action of a corresponding linear functional or covector from the dual space on the vector. For ket vectors there is a corresponding reversed notation $\langle\psi\rvert$ , called a bra, which represents the corresponding linear functional. Then the inner product between two states $\lvert\psi\rangle$ and $\lvert\phi\rangle$ can be written as

\langle\phi\mathclose{}|\mathopen{}\psi\rangle=-\langle\psi\mathclose{}|% \mathopen{}\phi\rangle.

(1.1)

This expression is known as a “bra-ket” (= bracket). This notation is due to Paul Dirac, and is known as Dirac notation or Bra-ket notation.

In a system where the Hilbert space is finite-dimensional with $n$ dimensions, it is isomorphic to $\mathbb{C}^{n}$ . Then the kets can be associated with column vectors and the bras with row vectors, just like how we would write linear functionals from the dual space in finite-dimensional linear algebra, and the inner product is defined by matrix multiplication. In this case, linear operators really are just $n\times n$ matrices.

All our familiar ideas from linear algebra carry over to Hilbert space, in particular if we have an operator $\hat{O}$ and it acts on the state $\lvert\psi\rangle$ to get

\hat{O}\lvert\psi\rangle=O\lvert\psi\rangle,

(1.2)

then this is an eigenvalue equation. We say that $\lvert\psi\rangle$ is an eigenket of $\hat{O}$ with eigenvalue $O$ .

There exists bases for the Hilbert space for which we can write the state $\lvert\psi\rangle$ as a linear combination. In particular there is a special basis which is formed from the solutions to the Schrodinger equation (see the next section) called the energy eigenbasis, which is usually labelled $\lvert n\rangle$ where $n$ is an integer label that can take some range of values specified by the problem. The expansion of an arbitrary state $\lvert\psi\rangle$ in the energy eigenbasis looks like

\lvert\psi\rangle=\sum_{n}c_{n}\lvert n\rangle=\sum_{n}\langle n\mathclose{}|% \mathopen{}\psi\rangle\lvert\psi\rangle,

(1.3)

, where $c_{n}$ is the inner product of the state $\lvert\psi\rangle$ with the eigenstate $\lvert n\rangle$ . We will study energy eigenbases again and again throughout the following chapters, as they are pretty much the most important objects of study in quantum mechanics.

1.1.2 The Wavefunction

For a system with spatial degrees of freedom, we also have a continuous basis that we can express our state in, the position basis $\lvert x\rangle$ . Since the position basis is continuous, an arbitrary quantum state $\psi$ is represented in the position basis by an integral:

\lvert\psi\rangle=\int_{-\infty}^{\infty}\psi(x,t)\lvert x\rangle\operatorname% {\mathrm{d}}\!x.

(1.4)

The quantity $\psi(x,t)$ is called the wavefunction, and is basically the continuous list of coefficients in the linear combination of position eigenkets.

We can isolate the value of the wavefunction at a position $x$ by calculating the inner product of the state $\lvert\psi\rangle$ with the relevant position eigenket $\hat{x}$ (taking care to use a different dummy variable for the integration):

$\displaystyle\langle x\mathclose{}\|\mathopen{}\psi\rangle$	$\displaystyle=\int_{-\infty}^{\infty}\langle x\rvert\psi(x^{\prime},t)\lvert x% ^{\prime}\rangle\operatorname{\mathrm{d}}\!x^{\prime}$	(1.5)
	$\displaystyle=\int_{-\infty}^{\infty}\psi(x^{\prime},t)\langle x\mathclose{}\|% \mathopen{}x^{\prime}\rangle\operatorname{\mathrm{d}}\!x^{\prime}$	(1.6)
	$\displaystyle=\int_{-\infty}^{\infty}\psi(x^{\prime},t)\delta(x-x^{\prime})% \operatorname{\mathrm{d}}\!x^{\prime}$	(1.7)
	$\displaystyle=\psi(x,t).$	(1.8)

For systems with only spatial degrees of freedom, which we will study exclusively in the first few chapters, we can actually get by using only the wavefunction to describe the full state of the system and ignoring the more abstract kets and bras. However, when we come to describe systems with internal degrees of freedom such as spin, we will need Dirac notation to describe the system state. To keep on top of things, we will keep track of ways that important concepts such as superpositions and expectation values can be described using both notations throughout.